<h2>题目编号 : 120</h2>
<div style="color:#666;font-size:80%;">21 April 2006</div><br />
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<p>Let <i>r</i> be the remainder when (<i>a</i><img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' />1)<img src="" style="display:none;" alt="^(" /><sup><i>n</i></sup><img src="" style="display:none;" alt=")" /> + (<i>a</i>+1)<img src="" style="display:none;" alt="^(" /><sup><i>n</i></sup><img src="" style="display:none;" alt=")" /> is divided by <i>a</i><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />.</p>
<p>For example, if <i>a</i> = 7 and <i>n</i> = 3, then <i>r</i> = 42: 6<img src="" style="display:none;" alt="^(" /><sup>3</sup><img src="" style="display:none;" alt=")" /> + 8<img src="" style="display:none;" alt="^(" /><sup>3</sup><img src="" style="display:none;" alt=")" /> = 728 <img src='images/symbol_cong.gif' width='9' height='11' alt='&equiv;' border='0' style='vertical-align:middle;' /> 42 mod 49. And as <i>n</i> varies, so too will <i>r</i>, but for <i>a</i> = 7 it turns out that <i>r</i><img src="" style="display:none;" alt="_(" /><sub>max</sub><img src="" style="display:none;" alt=")" /> = 42.</p>
<p>For 3 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <i>a</i> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 1000, find <span style='font-family:times new roman;font-size:13pt;'><img src='images/symbol_sum.gif' width='11' height='14' alt='&sum;' border='0' style='vertical-align:middle;' /></span> <i>r</i><img src="" style="display:none;" alt="_(" /><sub>max</sub><img src="" style="display:none;" alt=")" />.</p>

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